Hawking radiation in Reissner-Nordström blackhole with a global monopole via Covariant anomalies and Effective action
aa r X i v : . [ h e p - t h ] M a r Hawking radiation in Reissner-Nordstr¨om blackholewith a global monopole via Covariant anomalies andEffective action
Sunandan Gangopadhyay ∗ S. N. Bose National Centre for Basic Sciences,JD Block, Sector III, Salt Lake, Kolkata-700098, India
Abstract
We adopt the covariant anomaly cancellation method as well as the effective action approach to obtainthe Hawking radiation from the Reissner-Nordstr¨om blackhole with a global monopole falling in theclass of the most general spherically symmetric charged blackhole ( √− g = 1), using only covariantboundary condition at the event horizon.Keywords: Hawking radiation, Covariant anomaly, Effective action, Covariant Boundary condition PACS:
Introduction :
Hawking radiation is an important and prominent quantum effect arising from the quantization of matterfields in a background spacetime with an event horizon. The radiation is found to have a spectrum withPlanck distribution giving the blackholes one of its thermodynamic properties. Apart from the originalderivation by Hawking [1, 2], there is a tunneling picture [3, 4] based on pair creations of particles andantiparticles near the horizon and calculates WKB amplitudes for classically forbidden paths. A commonfeature in these derivations is the universality of the radiation: i.e. Hawking radiation is determineduniversally by the horizon properties (if we neglect the grey body factor induced by the effect of scatteringoutside the horizon).Recently, Robinson and Wilczek ([5]) proposed an interesting approach to derive Hawking radiation froma Schwarzschild-type black hole through gravitational anomaly. The method was soon extended to thecase of charged blackholes [6]. Further applications of this approach may be found in [7]-[13]. Thebasic idea in [5, 6] is that the effective theory near the horizon becomes two-dimensional and chiral.This chiral theory is anomalous. Using the form for two-dimensional consistent gauge/gravitationalanomaly, Hawking fluxes are obtained. However the boundary condition necessary to fix the parametersare obtained from a vanishing of covariant current and energy-momentum tensor at the horizon. Soonafter, the analysis of [5, 6] was reformulated in [14], [15] in terms of covariant expressions only. Thegeneralization of this approach to higher spin field has been done in [16].An alternative derivation of Hawking flux based on effective action using only covariant anomaly hasbeen discussed in [17], [18], [19]. This approach is particularly useful since only the exploitation of knownstructure of effective action near the horizon is sufficient to determine the Hawking flux. An importantingredient in this method is once again to realize that the effective theory near the event horizon becomestwo-dimensional and chiral. Another important aspect in this approach is the imposition of covariantboundary conditions only at the horizon.In this paper, we first adopt the covariant anomaly cancellation approach ([14]) to discuss Hawking ra-diation from Reissner-Nordstr¨om blackhole with a global monopole [21] which is an example of the mostgeneral spherically symmetric charged blackhole spacetime ( √− g = 1). Finally we adopt the effectiveaction approach [17] to reproduce the same result. However, as in [18], we shall once again ignore effectsto the Hawking flux due to scatterings by the gravitational potential, for example the greybody factor [20]. ∗ [email protected] awking radiation from Reissner-Nordstr¨om blackhole with a global monopole : The metric of a general non-extremal Reissner-Nordstr¨om blackhole with a global monopole O (3) is givenby [21] ds string = p ( r ) dt − h ( r ) dr − r d Ω (1)where, A = qr dt , p ( r ) = h ( r ) = 1 − η − mr + q r (2)with m being the mass parameter of the blackhole and η is related to the symmetry breaking scale whenthe global monopole is formed during the early universe soon after the Big-Bang [22]. The event horizonfor the above blackhole is situated at r H = (1 − η ) − (cid:2) m + p m − (1 − η ) q (cid:3) . (3)Now it has been argued in [11] that since the metric (1) is no longer asymptotically flat, so the well knownformula κ = 12 r − g rr g tt (cid:0) g tt, r (cid:1)(cid:12)(cid:12) r = r H ; g tt = p ( r ) , g rr = − h ( r ) (4)for computing the surface gravity for a general spherically symmetric asymptotically flat metric becomesproblematic to be applied in the case described by the metric (1) as it does not correspond to thenormalized time-like Killing vector. The correct surface gravity of the metric (1) is κ = 12 p − η p ′ ( r H ) (5)since it corresponds to the normalized time-like Killing vector l µ ( t ) = (1 − η ) − / ( ∂ t ) µ . (6)It is for this reason that the anomaly cancellation method as well as the effective action approach cannotbe immediately used to obtain the consistent formula of the Hawking temperature for the metric (1).Nevertheless, we can do the same analysis in another different way. By rescaling t → p − η t , we canrewrite the metric (1) as ds = f ( r ) dt − h ( r ) − dr + r d Ω f ( r ) = (1 − η ) h ( r ) , h ( r ) = 1 − η − mr − q r (7)and immediately derive the expected result for the Hawking temperature T = f ′ ( r H ) / (4 π p − η ).Hence, we shall apply the anomaly cancellation method and the effective action approach to the aboveform of the metric (7). The important point to note is that the determinant of the above metric √− g = 1. Anomaly cancellation approach :
With the aid of dimensional reduction procedure one can effectively describe a theory with a metric givenby the “ r − t ” sector of the full spacetime metric (7) near the horizon.Now we divide the spacetime into two regions and discuss the gauge/gravitational anomalies separately. Gauge anomaly
Since the spacetime has been divided into two regions, we divide the current J µ into two parts. Thecurrent outside the horizon denoted by J µ ( o ) is anomaly free and hence satisfies the conservation law ∇ µ J µ ( o ) = 0 . (8) Note that a spherically symmetric asymptotically bounded space-time metric without any loss of generality, can be castin the form ds = g tt dt + g rr dr + r d Ω . ∇ µ J µ ( H ) = − e π ¯ ǫ ρσ F ρσ = e π √− g ∂ r A t (9)where, ¯ ǫ µν = ǫ µν / √− g and ¯ ǫ µν = √− gǫ µν are two dimensional antisymmetric tensors for the upper andlower cases with ǫ tr = ǫ rt = 1.Now outside the horizon, the conservation equation (8) yields the differential equation ∂ r ( √− gJ r ( o ) ) = 0 (10)whereas in the region near the horizon, the anomaly equation (9) leads to the following differentialequation ∂ r (cid:16) √− gJ r ( H ) (cid:17) = e π ∂ r A t . (11)Solving (10) and (11) in the region outside and near the horizon, we get J r ( o ) ( r ) = c o √− g (12) J r ( H ) ( r ) = 1 √− g (cid:18) c H + e π Z rr H ∂ r A t ( r ) (cid:19) = 1 √− g ( c H + e π [ A t ( r ) − A t ( r H )]) (13)where, c o and c H are integration constants. Now as in [6], writing J r ( r ) as J r ( r ) = J r ( o ) ( r )Θ( r − r H − ǫ ) + J r ( H ) ( r ) H ( r ) (14)where, H ( r ) = 1 − θ ( r − r H − ǫ ), we find ∇ µ J µ = ∂ r J r ( r ) + ∂ r (ln √− g ) J r ( r )= 1 √− g ∂ r ( √− gJ r ( r ))= 1 √− g (cid:20)(cid:18) √− g ( J r ( o ) ( r ) − J r ( H ) ( r )) + e π A t ( r ) (cid:19) δ ( r − r + − ǫ ) + ∂ r (cid:18) e π A t ( r ) H ( r ) (cid:19)(cid:21) . (15)The term in the total derivative is cancelled by quantum effects of classically irrelevent ingoing modes.Hence the vanishing of the Ward identity under gauge transformation implies that the coefficient of thedelta function is zero, leading to the condition J r ( o ) ( r ) − J r ( H ) ( r )) + e π √− g A t ( r ) = 0 . (16)Substituting (12) and (13) in the above equation, we get c o = c H − e π A t ( r H ) . (17)The coefficient c H vanishes by requiring that the covariant current J r ( H ) ( r ) vanishes at the horizon. Hencethe charge flux corresponding to J r ( r ) is given by c o = √− gJ r ( o ) ( r ) = − e π A t ( r H ) = − e q πr H . (18)This is precisely the charge flux obtained in ([11]) using Robinson-Wilczek method of cancellation ofconsistent gauge anomaly. 3 ravitational anomaly In this case also, since the theory is free from anomaly in the region outside the horizon, hence we havethe energy-momentum tensor satisfying the conservation law ∇ µ T µ ( o ) ν = F µν J µ ( o ) . (19)However, the omission of the ingoing modes in the region r ∈ [ r + , ∞ ] near the horizon, leads to ananomaly in the energy-momentum tensor there. As we have mentioned earlier, in this paper we shallfocus only on the covariant form of d = 2 gravitational anomaly given by ([5, 6]): ∇ µ T µ ( H ) ν = F µν J µ ( H ) + 196 π ¯ ǫ νµ ∂ µ R = F µν J µ ( H ) + A ν . (20)It is easy to check that for the metric (1), the two dimensional Ricci scalar R is given by R = h f ′′ f + f ′ h ′ f − f ′ h f (21)and the anomaly is purely timelike with A r = 0 A t = 1 √− g ∂ r N rt (22)where, N rt = 196 π (cid:18) hf ′′ + f ′ h ′ − f ′ hf (cid:19) . (23)We now solve the above equations (19, 20) for the ν = t component. In the region outside the horizon,the conservation equation (19) yields the differential equation ∂ r ( √− gT r ( o ) t ) = √− gF rt J r ( o ) ( r ) = c o ∂ r A t (24)where we have used F rt = ∂ r A t and (12). Integrating the above equation leads to T r ( o ) t ( r ) = 1 √− g ( a o + c o A t ( r )) (25)where, a o is an integration constant. In the region near the horizon, the anomaly equation (20) leads tothe following differential equation ∂ r (cid:16) √− gT r ( H ) t (cid:17) = √− gF rt J r ( H ) ( r ) + ∂ r N rt ( r )= ( c H + e π [ A t ( r ) − A t ( r H )]) ∂ r A t ( r ) + ∂ r N rt ( r )= ∂ r (cid:18) e π (cid:20) A t ( r ) − A t ( r H ) A t ( r ) (cid:21) + N rt ( r ) (cid:19) (26)where we have used (13) in the second line and set c H = 0 in the last line of the above equation.Integration of the above equation leads to T r ( H ) t ( r ) = 1 √− g (cid:18) b H + Z rr H ∂ r (cid:18) e π (cid:20) A t ( r ) − A t ( r H ) A t ( r ) (cid:21) + N rt ( r ) (cid:19)(cid:19) = 1 √− g (cid:18) b H + e π [ A t ( r ) + A t ( r H )] − e π A t ( r H ) A t ( r ) + N rt ( r ) − N rt ( r H ) (cid:19) (27)where, b H is an integration constant.Writing the energy-momentum tensor as a sum of two contributions [6] T rt ( r ) = T r ( o ) t ( r ) θ ( r − r H − ǫ ) + T r ( H ) t ( r ) H ( r ) (28)4e find ∇ µ T µt = ∂ r T rt ( r ) + ∂ r (ln √− g ) T rt ( r )= 1 √− g ∂ r ( √− gT rt ( r ))= 1 √− g (cid:20) − e π A t ( r H ) ∂ r A t ( r ) + (cid:18) √− g ( T r ( o ) t ( r ) − T r ( H ) t ( r )) + e π A t ( r ) + N rt ( r ) (cid:19) δ ( r − r + − ǫ )+ ∂ r (cid:18) [ e π A t ( r ) + N rt ( r )] H ( r ) (cid:19)(cid:21) (29)where we have substituted the value of c from (18) in the last line.Now the first term in the above equation is a classical effect coming from the Lorentz force. The termin the total derivative is once again cancelled by quantum effects of classically irrelevant ingoing modes.The quantum effect to cancel this term is the Wess-Zumino term induced by the ingoing modes near thehorizon. Hence the vanishing of the Ward identity under diffeomorphism transformation implies that thecoefficient of the delta function in the above equation vanishes T r ( o ) t − T r ( H ) t + 1 √− g ( e π A t ( r ) + N rt ( r )) = 0 . (30)Substituting (25) and (27) in the above equation, we get a o = b H + e π A t ( r H ) − N rt ( r H ) . (31)The integration constant b H can be fixed by imposing that the covariant energy-momentum tensor van-ishes at the horizon. From (27), this gives b H = 0. Hence the total flux of the energy-momentum tensoris given by a o = e π A t ( r H ) − N rt ( r H )= e q πr H + 1192 π f ′ ( r H ) h ′ ( r H )= e q πr H + 1192 π f ′ ( r H )(1 − η ) . (32)This is precisely the Hawking flux obtained in ([11]) using Robinson-Wilczek method of cancellation ofconsistent anomaly. Effective action approach :
As we have already mentioned earlier, with the aid of dimensional reduction technique, the effective fieldtheory near the horizon becomes a two dimensional chiral theory with a metric given by the “ r − t ” sectorof the full spacetime metric (7) near the horizon.We now adopt the methodology in [17]. For a two dimensional theory the expressions for the anomalous(chiral) and normal effective actions are known [23]. We shall use only the anomalous form of the effectiveaction for deriving the charge and the energy flux. The current and the energy-momentum tensor in theregion near the horizon is computed by taking appropriate functional derivative of the chiral effectiveaction. Next, the parameters appearing in the solution is fixed by imposing the vanishing of covariantcurrent and energy-momentum tensor at the horizon. Once these are fixed, the charge and the energyflux are obtained by taking the asymptotic ( r → ∞ ) limit of the chiral current and energy-momentumtensors. We also use the expression for the normal effective action to establish a connection between thechiral and the normal current and energy-momentum tensors.With the above methodology in mind, we write down the anomalous (chiral) effective action (describingthe theory near the horizon) [23] Γ ( H ) = − z ( ω ) + z ( A ) (33)5here A µ and ω µ are the gauge field and the spin connection and z ( v ) = 14 π Z d x d y ǫ µν ∂ µ v ν ( x )∆ − g ( x, y ) ∂ ρ [( ǫ ρσ + √− gg ρσ ) v σ ( y )] (34)where ∆ g = ∇ µ ∇ µ is the laplacian in this background.The energy-momentum tensor is computed from a variation of this effective action. To get their covariantforms in which we are interested, one needs to add appropriate local polynomials [23]. Here we quote thefinal result for the chiral covariant energy-momentum tensor and the chiral covariant current [23]: T µν = e π D µ BD ν B + 14 π (cid:18) D µ GD ν G − D µ D ν G + 124 δ µν R (cid:19) (35) J µ = − e π D µ B (36)where D µ is the chiral covariant derivative D µ = ∇ µ − ¯ ǫ µν ∇ ν = − ¯ ǫ µν D ν . (37)Also B ( x ) and G ( x ) are given by B ( x ) = Z d y √− g ∆ − g ( x, y )¯ ǫ µν ∂ µ A ν ( y ) (38) G ( x ) = Z d y ∆ − g ( x, y ) √− g R ( y ) (39)and satisfy ∇ µ ∇ µ B = − ∂ r A t ( r ) , ∇ µ ∇ µ G = R . (40)The solutions for B and G read B = B o ( r ) − at + b ; ∂ r B o = 1 √ f h ( A t ( r ) + c ) (41) G = G o ( r ) − pt + q ; ∂ r G o = − √ f h s hf f ′ + z ! (42)where a, b, c, p, q, z are constants of integration.By taking the covariant divergence of (35) and (36), we get the anomalous Ward identities (20) and (9).In the region away from the horizon, the effective theory is given by the standard effective action Γ of aconformal field with a central charge c = 1 in this blackhole background [23] and reads:Γ = 196 π Z d xd y √− g R ( x ) 1∆ g ( x, y ) √− g R ( y ) + e π Z d xd yǫ µν ∂ µ A ν ( x ) 1∆ g ( x, y ) ǫ ρσ ∂ ρ A σ ( y ) . (43)The covariant energy-momentum tensor T µν ( o ) and the covarant gauge current J µ ( o ) in the region outsidethe horizon are given by T µν ( o ) = 2 √− g δ Γ δg µν = 148 π (cid:18) g µν R − ∇ µ ∇ ν G + ∇ µ G ∇ ν G − g µν ∇ ρ G ∇ ρ G (cid:19) + e π (cid:18) ∇ µ B ∇ ν B − g µν ∇ ρ B ∇ ρ B (cid:19) (44) J µ ( o ) = δ Γ δA µ = e π ¯ ǫ µν ∂ ν B (45)6nd satisfy the normal Ward identities (19) and (8). Charge and Energy Flux:
In this section we calculate the charge and the energy flux by using the expressions for the anomalouscovariant gauge current (36) and anomalous covariant energy-momentum tensor (35). We will show thatthe results are the same as that obtained by the covariant anomaly cancellation method.Using the solution for B ( x ) (38) and (37), the µ = r component of the anomalous (chiral) covariant gaugecurrent (36) becomes J r ( r ) = e π s hf [ A t ( r ) + a + c ] . (46)Now implementation of the boundary condition namely the vanishing of the anomalous (chiral) covariantgauge current at the horizon, J r ( r H ) = 0, leads to a + c = − A t ( r H ) . (47)Hence the expression J r ( r ) reads J r ( r ) = e π s hf [ A t ( r ) − A t ( r H )] . (48)Now the charge flux is given by the asymptotic ( r → ∞ ) limit of the anomaly free covariant gaugecurrent (45). Now from (9), we observe that the anomaly vanishes in this limit. Hence the charge flux isabstracted by taking the asymptotic limit of the above equation multiplied by an overall factor of √− g .This yields c = ( √− gJ r )( r → ∞ ) = ( r fh J r )( r → ∞ ) = − e π A t ( r H ) = − e q πr H (49)which agrees with (18).We now consider the normal (anomaly free) covariant gauge current (45) to establish its relation withthe chiral (anomalous) covariant gauge current (48). The µ = r component of J µ ( o ) is given by J r ( o ) ( r ) = e π s hf a . (50)The asymptotic form of the above equation (50) must agree with the asymptotic form of (46) . Thisyields (using (47)): a = c = − A t ( r H ) . (51)Using the above solutions in (46) and (50) yields (48) and √− gJ r ( o ) ( r ) = r fh J r ( o ) ( r ) = − e π A t ( r H ) . (52)The above expressions (48) and (52) yields the equation between the chiral (anomalous) and the normalenergy-momentum tensors (16).Now we focus our attention on the gravity sector. Using the solutions for B ( x ) (38) and G ( x ) (39), the r − t component of the anomalous (chiral) covariant energy-momentum tensor (35) becomes T rt ( r ) = e π s hf [ A t ( r ) − A t ( r H )] + 112 π s hf " p − s hf f ′ + z ! + 124 π s hf "s hf f ′ p − s hf f ′ + z !! + 14 hf ′′ − f ′ (cid:18) hf f ′ − h ′ (cid:19) . (53) This is true since the anomaly in the asymptotic limit ( r → ∞ ) vanishes as can be readily seen from (9). T rt ( r H ) = 0, leads to p = 14 h z ± p f ′ ( r H ) h ′ ( r H ) i ; f ′ ( r H ) ≡ f ′ ( r = r H ) . (54)Using either of the above solutions in (53) yields T rt ( r ) = e π s hf [ A t ( r ) − A t ( r H )] + 1192 π s hf (cid:20) f ′ ( r H ) h ′ ( r H ) − hf f ′ + 2 hf ′′ + f ′ h ′ (cid:21) . (55)The energy flux is now given by the asymptotic ( r → ∞ ) limit of the anomaly free energy-momentumtensor (44). Now from (20), we observe that the anomaly vanishes in this limit. Hence the energy flux isabstracted by taking the asymptotic limit of the above equation multiplied by an overall factor of √− g .This yields a = ( √− gT rt )( r → ∞ ) = ( r fh T rt )( r → ∞ )= e π A t ( r H )] + 1192 π f ′ ( r H ) h ′ ( r H )= e q πr H + 1192 π f ′ ( r H ) h ′ ( r H ) (56)which correctly reproduces the energy flux (32).We now consider the normal (anomaly free) energy-momentum tensor (44) to establish its relation withthe chiral (anomalous) energy-momentum tensor (55). The r − t component of T µν ( o ) is given by T rt ( o ) ( r ) = e π s hf a [ A t ( r ) + c ] − π s hf zp = − e π s hf A t ( r H )[ A t ( r ) − A t ( r H )] − π s hf zp (57)where we have used (51). Once again since the anomaly in the asymptotic limit ( r → ∞ ) vanishes as canbe readily seen from (20), the asymptotic form of the above equation (57) must agree with the asymptoticform of (53). This yields: p = − z . (58)Solving (54) and (58) gives two solutions for p and z : p = 18 p f ′ ( r H ) h ′ ( r H ) ; z = − p f ′ ( r H ) h ′ ( r H ) p = − p f ′ ( r H ) h ′ ( r H ) ; z = 12 p f ′ ( r H ) h ′ ( r H ) . (59)Using either of the above solutions in (53) and (57) yields (55) and √− gT rt ( o ) ( r ) = r fh T rt ( o ) ( r ) = − e π A t ( r H )[ A t ( r ) − A t ( r H )] + 1192 π f ′ ( r H ) h ′ ( r H ) . (60)The above expressions (55) and (60) yields the equation between the chiral (anomalous) and the normalenergy-momentum tensors (30). Discussions :
In this paper, we studied the problem of Hawking radiation from Reissner-Nordstr¨om blackhole witha global monopole using covariant anomaly cancellation technique and effective action approach. Thepoint to note in the anomaly cancellation method is that Hawking radiation plays the role of cancelling8auge and gravitational anomalies at the horizon to restore the gauge/diffeomorphism symmetry at thehorizon. An advantage of this method is that neither the consistent anomaly nor the counterterm relatingthe different (covariant and consistent) currents, which were essential ingredients in [6], were required.On the other hand, in the effective action technique, we only need covariant boundary conditions, theimportance of which was first stressed in [17]. Another important input in the entire procedure is theexpression for the anomalous (chiral) effective action (which yields anomalous Ward identity havingcovariant gauge/gravitational anomaly). The unknown parameters in the covariant current and energy-momentum tensor derived from this anomalous effective action were fixed by a boundary condition-namely the vanishing of the covariant current and energy-momentum tensor at the event horizon of theblackhole. Finally, the charge and the energy flux were extracted by taking the r → ∞ limit of the chiralcovariant current and energy-momentum tensor. The relation between the chiral and the normal currentand energy-momentum tensors is also established by requiring that both of them match in the asymptoticlimit which is possible since the gauge/gravitational anomaly vanishes in this limit. Acknowledgements
The author would like to thank Prof. R. Banerjee and Shailesh Kulkarni for useful comments.
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